Gaussian curvature

From Academic Kids

Gaussian curvature of a point on a surface is the product of the principal curvatures, k1 and k2 of the given point.

Symbolically, the Gaussian curvature K is defined as

<math> K = k_1 k_2 <math>.

It is also given by

<math>K = \frac{\langle (\nabla_2 \nabla_1 - \nabla_1 \nabla_2)\mathbf{e}_1, \mathbf{e}_2\rangle}{\det g}<math>

where <math>\nabla_i = \nabla_{{\mathbf e}_i}<math> is the covariant derivative and g is the metric tensor.

At a point p on a regular surface in <math>\mathbb{R}^3<math>, the Gaussian curvature is also given by

<math>K(\mathbf{p}) = \det(S(\mathbf{p}))<math>

where S is the shape operator.

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